| Abstract: |
| The fully parabolic quasilinear Keller--Segel system
\[
\begin{cases}
u_t = \nabla \cdot ((u+1)^{m-1} \nabla u - u(u+1)^{q-1} \nabla v), \
v_t = \Delta v - v + u,
\end{cases}
\]
which we consider in a ball $\Omega \subset \mathbb R^n$, $n \ge 2$,
admits unbounded solutions whenever $m, q \in \mathbb R$ satisfy $m - q < \frac{n-2}{n}$.
These are necessarily global in time if $q \leq 0$ and finite-time blow-up is known to be possible if $q > 0$ and $\max\{m, q\} \geq 1$.
Utilizing certain pointwise upper estimates for $u$, we are able to give an affirmative answer to the (for nearly a decade formerly open) question whether solutions may blow up in finite time if $\max\{m, q\} < 1$.
If $n = 2$, for instance, we construct solutions blowing up in finite time whenever ($m-q < 0$ and) $q < 2m$. |
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